Co-Coatomically Supplemented Modules

Abstract

It is shown that if a submodule N of M is co-coatomically supplemented and M/N has no maximal submodule then M is a co-coatomically supplemented module. If a module M is co-coatomically supplemented then every finitely M-generated module is a co-coatomically supplemented module. Every left R-module is co-coatomically supplemented if and only if the ring R is left perfect. Over a discrete valuation ring a module M is co-coatomically supplemented if and only if the basic submodule of M is coatomic. Over a nonlocal Dedekind domain if the torsion part T(M) of a reduced module M has a weak supplement in M then M is co-coatomically supplemented if and only if M/T (M) is divisible and T (P) (M) is bounded for each maximal ideal P. Over a nonlocal Dedekind domain if a reduced module M is co-coatomically amply supplemented then M/T (M) is divisible and T (P) (M) is bounded for each maximal ideal P. Conversely if M/T (M) is divisible and T (P) (M) is bounded for each maximal ideal P then M is a co-coatomically supplemented module.